I think the USA consists of something like 4 or 5% Asians, but a lot of the law schools consist of double or triple that number. You can break it down by Japanese, Korean, Chinese, Indian, et al but the bottom line is still the same: no matter how you cut it, Asians are over-represented at most law schools.
Now maybe if you apply to a law school in some part of the country that doesn't have a lot of whatever type of Asian you are (assuming you are), that in fact has less than the national percentage for your type, then it's possible, you might be considered an URM for their school.
Also, if you are a Southeast Asian, for example, from Vietnam or Laos or Cambodia, or one of those places, then you might very well be considered an URM because there are not many of SEAsians in law school. Most SEAsians suffered a lot, too, during the 1960s, and had to escape their war-torn lands. So if you come from that background, then you may indeed be considered URM.
But URM status is not limited by ethnicity, necessarily (though for obvious reasons it does tend to fall along ethnic lines). If you have a compelling story to tell, something demonstrably difficult that most applicants from mainstream America have never had to face, then that might carry just as much weight with an adcomm as does URM status.